Think 2-D Is Cool? Wait Till You Try 3-D!

For most people, they will think "The side length must be from a lattice point, to another lattice point. Hence, it must be of the form $\sqrt{x^2 + y^2 + z^2 }$ ." This leads them to choose $\{ n \mid n \text{ is the sum of 3 perfect squares.} \}$ , which is not correct.

As this problem is on Construction , we need to show that such a lattice cube with those side lengths actually exists. This may not always be true. For example, let's show that there is no lattice cube with side length $\sqrt{3} = \sqrt{1 + 1 + 1 }$ .

Proof: WLOG, one of the vertices is at $(0,0,0)$ . Then, the 3 vertices that are connected to it, are of the form $(\pm 1, \pm 1, \pm 1 )$ . Observe that the distance between any 2 of these vertices is either 2, $\sqrt{8}$ or $\sqrt{12}$ . However, if they were the vertices of a cube of side length $\sqrt{3}$ , then the distance must be $\sqrt{3} \times \sqrt{2}$ . Hence, we have a contradiction. $_ \square$

This shows us that the question is actually more complex and subtle. There are several ways to proceed, and I'd present a proof using vectors, as that is slightly more approachable than the proof using matrices.

Proof (general case): Let the cube have side length $\sqrt{k}$ . (Why must it be in this form?) Let one of the vertices be $(0,0,0)$ . Let the 3 adjacent vertices be of the form $(a_x, a_y, a_x), (b_x, b_y, b_z), (c_x, c_y, c_z)$ . Then, we know that

$\begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix} \times \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix} = \pm \sqrt{k} \cdot \begin{pmatrix} c_x \\ c_y \\ c_z \end{pmatrix}$

Since the LHS are all integers, hence the RHS are also all integers. This implies that $\sqrt{k}$ must be an integer.

Conversely, for any integer $n$ , there is clearly a lattice cube of side length $n$ . Hence, the set of side lengths of a lattice cube is $\{ n \mid n \text{ is an integer.} \}$ . $_ \square$

since the edges have to be integral, a non integral length for the side of the cube would result into at least one of the edges to have a non-integral co-ordinate... Hence, the sides have to be a positive integer...